Special Math Department Colloquium: Vincent Bouchard, University of Alberta

Feynman's famous quote is of particular relevance for research at the interface of mathematics and physics in recent decades. A striking example is the impressive number of fascinating results (many of them still conjectural) in various areas of mathematics, such as geometry, topology and number theory, that have been obtained via string theory. In this vein, mirror symmetry and topological string theory have been particularly fruitful. In this talk I will focus on a number of mathematical conjectures and theorems that we have obtained through careful study of mirror symmetry. I will discuss what string theory tells us about (quasi-)modularity of the generating functions of Gromov-Witten invariants of Calabi-Yau threefolds, and what it implies for the crepant resolution conjecture relating Gromov-Witten invariants of an orbifold to the invariants of its crepant resolution. I will also talk about a new mysterious recursive structure conjecturally satisfied by the Gromov-Witten generating functions for toric Calabi-Yau threefolds, with far-reaching and still mostly unexplored consequences. By the end of the talk, you should hopefully be convinced of "the unreasonable effectiveness of string theory in mathematics"!